Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

Rietsch, Konstanze

doi:10.1090/S0894-0347-02-00412-5

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ISSN 1088-6834(online) ISSN 0894-0347(print)

Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

Author: Konstanze Rietsch
Journal: J. Amer. Math. Soc. 16 (2003), 363-392
MSC (2000): Primary 20G20, 15A48, 14N35, 14N15
Posted: November 29, 2002
Erratum: J. Amer. Math. Soc. 21 (2008), 611-614
MathSciNet review: 1949164
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Abstract: We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in forms a real semi-algebraic cell of dimension . Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\mathbb{C} )$ relying in particular on the positivity of the structure constants, which are enumerative Gromov-Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson's which we explain with proofs in the type case.

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